Limit of ln(sinx) & Limit of [ln(1+x^2)-ln(1+x)]

[QuarkCharmer]

Homework Statement

$lim_{x \to 0+} ln(sin(x))$

$lim_{x \to \infty} [ln(1+x^2)-ln(1+x)]$

Homework Equations

The Attempt at a Solution

I'm really not sure how to take this limit at all?

I know (from using a table) that it tends towards -infinity, but I am not sure how to go about taking it manually?

I am thinking that because as x approaches 0, sin(x) approaches 0, you can treat sin(x) like x. Then as x approaches 0, ln(x) approaches -infinity.

For the second one, if I separate it into two limits, they both go towards infinity, then it's infinity - infinity?

 

[LCKurtz]

Homework Statement

$lim_{x \to 0+} ln(sin(x))$

Homework Equations

The Attempt at a Solution

I'm really not sure how to take this limit at all?

I know (from using a table) that it tends towards -infinity, but I am not sure how to go about taking it manually?

I am thinking that because as x approaches 0, sin(x) approaches 0, you can treat sin(x) like x. Then as x approaches 0, ln(x) approaches -infinity.

Intuitively, that is correct. You can make it rigorous with a δ, ε argument. If you can use the fact that ln(x) → -∞ as x → 0⁺ then you know that given any N > 0 there is a δ > 0 such that ln(x) < -N if 0 < x < δ. Now put that together with a similar type of argument about sin(x), knowing that sin(x) → 0 as x → 0.

 

[micromass]

For the first one, I think the most rigorous approach (besides epsilon-delta) is the squeeze theorem. Use

$$\sin(x)\leq x$$

Take logs and take the limit.

For the second one, first make one logarithm using

$$ln(a/b)=ln(a)-ln(b)$$